Hint: define the map by show that is a well-defined surjective ring homomorphism and:

January 16th 2009, 05:35 AM

Inti

thereīs one thing thatīs not clear to me:
Iīve just proved that , which means that is isomorphic to , which is a field.
That allows me to say that the ideal is prime?
As far as Iīm concerned, the quotient of a ring by a prime ideal is a domain.
Iīm kind of confused with that...
But the help was very helpful (Nod)

And I have one more question: is the product of two ideals the set of all the products of the elements of each ideal??

Thanks!

January 16th 2009, 05:55 PM

NonCommAlg

Quote:

Originally Posted by Inti

thereīs one thing thatīs not clear to me:
Iīve just proved that , which means that is isomorphic to , which is a field.
That allows me to say that the ideal is prime?

that means the ideal is maximal and we know that every maximal ideal is prime.

Quote:

And I have one more question: is the product of two ideals the set of all the products of the elements of each ideal??

no. if and are two ideals, then however, for example, if you want to prove that where is an ideal, then you only need to prove that